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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 88935p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
88935.j3 | 88935p1 | \([1, 1, 1, -14946, 660078]\) | \(1771561/105\) | \(21884349909345\) | \([2]\) | \(276480\) | \(1.3128\) | \(\Gamma_0(N)\)-optimal |
88935.j2 | 88935p2 | \([1, 1, 1, -44591, -2814316]\) | \(47045881/11025\) | \(2297856740481225\) | \([2, 2]\) | \(552960\) | \(1.6594\) | |
88935.j4 | 88935p3 | \([1, 1, 1, 103634, -17399656]\) | \(590589719/972405\) | \(-202670964510444045\) | \([2]\) | \(1105920\) | \(2.0059\) | |
88935.j1 | 88935p4 | \([1, 1, 1, -667136, -209997292]\) | \(157551496201/13125\) | \(2735543738668125\) | \([2]\) | \(1105920\) | \(2.0059\) |
Rank
sage: E.rank()
The elliptic curves in class 88935p have rank \(1\).
Complex multiplication
The elliptic curves in class 88935p do not have complex multiplication.Modular form 88935.2.a.p
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.